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Y. Yuan#, S. Wang, N. Wang, S. Xu, IHEP, Beijing, 100049, P.R.C.AbstractA new code for longitudinal beam dynamics design andbeam simulation in proton synchrotron has beendeveloped. In this code, the longitudinal beam dynamicsdesign can be performed for arbitrary curve of dipolemagnetic field, and for both basic harmonic cavity anddual harmonic cavity. The beam dynamics simulationwith space charge effect can be done in longitudinal phasespace, also for both basic harmonic cavity and dualharmonic cavity. The influence of stray fields of RFcavity, which is the higher order mode of cavity comingfrom the RF generator, on the beam can also be simulatedby using the code.INTRODUCTIONIn the design of Rapid Cycling Synchrotron (RCS) ofChina Spallation Neutron Source (CSNS/RCS) [1] [2],RAMA and ORBIT [3] are used for longitudinal beamdynamics design and beam dynamics simulation.However, RAMA does not work for the dipole fieldramping deviated from sinusoidal curve, and also can’tperform the longitudinal beam dynamics design with dualharmonic cavity. ORBIT can not perform the simulationwith dual harmonic cavity, or with a dipole field rampingdeviated from sinusoidal curve. To meet the requirementof beam dynamics design and study in CSNS/RCS, thecode C-SCSIM was developed. The longitudinal beamdynamics design can be performed for arbitrary curve ofdipole magnetic field, and for both basic harmonic cavityand dual harmonic cavity. The beam dynamics simulationwith space charge effect can be done in longitudinal phasespace, also for both basic harmonic cavity and dualharmonic cavity. The influence of stray fields of RFcavity, which is the higher order mode of cavity comingfrom the RF generator, on the beam, can also besimulated by using the code. The key issues on the codedevelopment are given, and the results of longitudinalbeam dynamics design and beam simulation are alsopresented. Figure 1 shows the issues considered in thecode.

Figure 1: The simulation elements in C-SCSIM.BEAM DYNAMICS DESIGN WITH DUALHARMONIC RF SYSTEMMany proton synchrotrons are designed to use dualharmonic RF system to increase the bunching factor, aswell as to decrease the transverse space charge effect. Asa longitudinal beam dynamics tracking code, C-SCSIMcan meet the requirement of physical design in dualharmonic RF system. The physical mechanism andprogram algorithm for simulation with dual RF system inC-SCSIM has been strictly tested and checked. With dualharmonic RF system, the basic longitudinal beamdynamics can be expressed as:

( ) ( )1 2 2( )sin sin(2 ),i idB tV V Ldt     (1)where  is the bend radius of the dipoles, L is thecircumference and B(t) is the magnetic strength of thedipole magnet. (i)is the accelerating phase for the ithparticle and 2is phase advance of the second harmonicas shown in Fig. 3.On the other hand, when dual RF system is introduced,the “potential energy” contributed by the secondharmonic should be included. The synchrotron motion ofparticles becomes [4],

ORBIT in the same initial condition. In order to improvethe bunching factor as large as possible, simulationexperiment has been done with many different series ofsecond harmonic voltages and phases under the conditionof larger bunching factor. Finally, an optimized result ofRF voltages has been found, which is shown in Fig. 5.

Figure 3: bunching factor for CSNS/RCS for single RFsystem and dual RF system.

Figure 4: bunching factor for CSNS/RCS compared withORBIT.Figure 5: RF voltage optimized for dual RF system inCSNS/RCS.LONGITUDINAL SPACE CHARGESIMULATION BASED ON FFTSpace charge effect is an important issue for protonsynchrotron, especially when running with high beamintensity. For computer simulation, the method ofParticle-In-Cell (PIC) is widely used in some of present 3-D tracking codes, with which the space charge effect ofthe beam is evaluated by calculating the coulomb force ofeach “finite size particle” in the transverse direction.However, a common space charge electromagnetic fieldcan be used to describe the space charge force in thelongitudinal direction, instead of the calculation of theforce acting on each particle [5]. Using this method, theinfluence of longitudinal space charge impedance andwall coupling impedance on the beam can be simulatedby the tracking code. The aspect of longitudinal spacecharge effect in C-SCSIM is developed under thismechanism. See Fig. 6.

Figure 6: The distribution of the electromagnetic fields ofbeam in the pipe.The energy of ithparticle acquired from the commonlongitudinal space charge electromagnetic field can bedescribed as

0 02arctan( )2n WnhZ gZ  (4)The FFT method is used in the code to calculate thereal part and imaginary part of beam current so that theseries of amplitude and phase of beam current can beobtained through transforming them. Besides, the seriesof amplitude is needed to be normalized so that the firstterm gives the correct average beam current.It is valuable to point out how to choose the rightsampling time to ensure the precision of the resolution ofthe frequency after FFT. In C-SCSIM, in which the 2-based FFT is realized, take tsas the sampling time, where

cLts2

and100Fftfs,

where f0and fsis the revolution frequency and samplingfrequency separately.In the code, 2nbins are taken averagely in the range of2π in longitudinal phase space. A series of 2n-1values ofamplitude in frequency domain can be generated throughFFT.Two of samples are in Figure 7, showing the resultswith and without the longitudinal space charge at 0.5015ms when injection process has just finished in J-PARC/RCS.

(a) (b)

Figure 7: Particle tracking (a) without space charge, (b)with at 308thturn with space charge.Besides, if the real part and imaginary part of eachharmonic of some kind of longitudinal impedance isgiven, particles can be tracked to simulate the influence ofthe longitudinal impedance on the beam.BEAM SIMULATION WITH STRAYFIELD IN FERRITE LOADED RF CAVITYCoaxial cavities are often usedin this kind of protonsynchrotron. As to the rapid cycling synchrotron (RCS),ferrite-loaded cavity is needed to synchronize theresonance frequencies to the revolution frequencies.There often exist many stray fields besides thefundamental field only whichis used to accelerate. Theinfluence of these stray fields to the beam behaviour isvaluable to study and simulate by computer programbecause the stray fields canprobably affect the beamstrongly in the condition the stray fields resonate with thesynchrotron sideband.The equation of synchrotron motion for the stray fieldelement is:

122[sin( ) sin( )]n n m m s mmeV m mE     (5)where thenrepresent the nth particle, the m is the orderof the stray fields.ΦsandΦmare the synchrotron phaseand the stray field phase respectively. Treating the strayfields in the RF cavity as “another RF cavity”, thesimulation code, C-SCSIM, can evaluate reasonably theinfluence of these stray fields on the beam. Just like themethod of calculating the space charge effects, theparticles experience a “small cavity” on behalf of theinfluence of the stray fieldsbesides the ideal acceleratingcavity. See Figure 1.These stray fields in RF cavities can be excited by theRF power supply which is not a pure signal and can beseen by the spectrum analyzer. The values of thefrequency and amplitude of each order of stray fields canbe acquired by FFT from the initial data output fromoscilloscope in the experiment.Some simulations have been done to evaluate theinfluence of the stray fields on the beam and optimize theRF cavity for CSNS/RCS.As is shown in Table 1, there exists one busbar mode,which could be resonated with some order of stray field,can not be ignored.

Table 1:Comparison Before and After OptimizationBeforeAfterResonance time(ms)14-16 7.5-8.5Resonancefrequency(MHz)7.05 7.75Resonance order1.1856-1.2073 0.9599-1.0185Maximal Amplitude1/5.6 1/13Phasevarying varyingAfter optimization, the resonance order has changedfrom 3rdto 4th. The result from the simulation shows thatthe beam loss is much less.The shape of the bunch seemsto more regular than that before optimization, see Fig. 8.CONCLUSIONBased on the longitudinal physical model andreasonable algorithm, the code C-SCSIM is a newparticle-tracking code whose simulation results have beenchecked strictly and compared with other world wideused tracking code. So it can be used for protonsynchrotron design and longitudinal parameteroptimization. The functions comprises of the basicparticle tracking in the given proper voltages orsynchrotron phase, the dual RF system simulation,longitudinal space charge effects and stray fieldsoptimization. C-SCSIM has been already used in thedesign of the CSNS/RCS.ACKNOWLEDGEMENTSThe author would like to thank CSNS/RCS RF group inIHEP for measurement of thestray field in RF cavity, Y.An, J. Qiu for their helpful discussion.REFERENCES[1] CSNS Feasibility Study Report, June, 2009, IHEP[2] S. Wang, THE OPTIMIZATION OF BEAMDYNAMICS DESIGN FORCSNS/RCS, IPAC 2010[3] J.D.Galambos et al. ORBIT User Manual, 1999[4] S.Y.Lee, Accelerator Physics, second edition, (WorldScientific,Singapore,